Equity Research puzzles, solved step by step

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All topicsProbability and brainteasers12Expected value and decisions8Market sizing and estimation12Returns and compounding9Valuation riddles12Three statement riddles10EPS and share count9Cost of capital and rates8Growth, mix and unit economics8Mental maths6Data and reasoning traps6
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Showing 11–20 of 100
  1. 011A company has two segments. This year both segments improved their operating margins, yet the group's operating margin fell. Give numbers that make this true, and explain what happened.Data and reasoning trapsCoreSell-side equity researchResearch KPO and GCC

    Try it first

    Before building the example: what has to change for this to happen?

    Show the worked solution

    The revenue mix shifted towards the lower-margin segment. Say segment A earns 30% on 80 of revenue and B earns 10% on 20: the group earns 26%. Next year A earns 32% on 40 and B earns 12% on 60. Both improved by 2 points, but the group earns 20 on 100, 20%, because most revenue now sits in the 12% business.

    How can an average fall when every part of it rises?

    Think of a class where both the science and arts sections improve their average marks, yet the school's overall average falls, because far more students joined the lower-scoring section this year. An overall average depends on the weights as well as the parts, so a big enough shift in weights can reverse the direction of the average. Statisticians call this Simpson's paradoxThe pattern where a trend that holds in every group reverses when the groups are combined, because the group sizes changed..

    Both segments improve, the group gets worse: the weights moved0%10%20%30%30%Segment A 32%10%Segment B 12%26%Group 20%Year 1Year 2Revenue mixA 80%B 20%Year 1A 40%B 60%Year 2: mix shifts to the 12% segmentGroup margin is aweighted average.Weights moved from80/20 to 40/60,so the average fell6 points.
    Segment A's margin rises from 30% to 32% and segment B's from 10% to 12%, but revenue moves from 80/20 to 40/60 in favour of B, so the group margin falls from 26% to 20%.

    How do you show the numbers work?

    SegmentYear 1 revenueYear 1 marginYear 2 revenueYear 2 margin
    A8030%4032%
    B2010%6012%
    Group26%20%
    Group profit falls from 26 to 20 on flat revenue of 100, even though both segment margins rise by two points.

    Check the group line each year. Year 1: 80 x 30% plus 20 x 10% is 24 plus 2, or 26 on 100. Year 2: 40 x 32% plus 60 x 12% is 12.8 plus 7.2, or 20 on 100. The whole fall comes from moving 40 of revenue out of a 30% business and into a 10% one, which costs 8 points of group margin; the segment improvements claw back only 2.

    In a research note this is the difference between a margin miss that signals trouble and one that simply reflects mix. Split the change into a mix effect and a rate effect before judging management. The limitation: the split depends on which year's weights you use, so state the convention.

    Where candidates lose it

    The common loss is saying it cannot happen, or inventing a hidden cost to explain it. The interviewer wants to hear the words weighted average and mix within the first sentence.

    The second is giving the concept without numbers. The question asks for an example; build a two-row table on the spot and check the group line out loud.

    What the interviewer asks next

    • Split the 6 point fall into a mix effect and a margin effect.
    • Where have you seen mix drive a company's reported margin in results?
    • Can revenue growth fall while every segment grows faster than before?
  2. 012You have a coin that lands heads with probability 1/3. How do you use it to produce a fair 50/50 result, and on average how many flips of the biased coin does each fair result cost?Probability and brainteasersCoreD.E. ShawNew York · 2026

    Try it first

    You flip twice and keep only HT or TH. On average, how many biased flips per fair result?

    Show the worked solution

    Flip twice: call heads on HT, tails on TH, and flip two more times on HH or TT. HT and TH each have probability 1/3 x 2/3 = 2/9, so they are equally likely whatever the bias. A pair is kept with chance 4/9, so on average you need 9/4 pairs, which is 4.5 flips per fair result.

    Why are HT and TH always equally likely?

    Think of two friends who each toss the same lopsided coin once. The chance the first gets heads and the second tails is the same as the chance the first gets tails and the second heads, because the same two numbers are multiplied, just in a different order. For any bias p, HT and TH both have probability p times (1 minus p), so the order of two different outcomes is a perfectly fair coin. This is known as the von Neumann trick.

    Two flips of a 1/3 coin: HT and TH have equal area whatever the biasHH1/9discardTH2/9call tailsHT2/9call headsTT4/9discardH: 1/3T: 2/3First flipHTSecond flipKeep only the mixed pairsH then T: 1/3 x 2/3 = 2/9T then H: 2/3 x 1/3 = 2/9Same product, different order: always equalCost of one fair flipA pair is kept with chance 4/9Expected pairs: 9/4 = 2.25Expected biased flips: 4.5Theoretical best with this coin: about 1.09flips per fair result (0.918 fair bits per flip)
    With a one-third heads coin, two flips give HH with probability 1/9, HT and TH with 2/9 each and TT with 4/9, so keeping only HT and TH gives a fair result and needs 4.5 biased flips on average.

    How do you get the 4.5 flips, and can you do better?

    Each pair succeeds with chance 4/9. The number of pairs until the first success averages one over the success chance, 9/4, and each pair is two flips. So the fair coin costs 4.5 biased flips on average, a steep price for throwing away five pairs in every nine.

    The relationship
    E[flips]=2×12p(1−p)=2×94=4.5E[\text{flips}] = 2 \times \frac{1}{2p(1-p)} = 2 \times \frac{9}{4} = 4.5
    pthe chance of heads, 1/3
    2p(1-p)the chance a pair is HT or TH, 4/9
    What it says in wordsTwo flips per attempt, divided by the chance an attempt succeeds.

    The interviewer's follow-up is efficiency. The discarded pairs still carry information: HH and TT are themselves a biased coin, and you can run the same trick on them, and on the pattern of which pairs matched. Iterating like this recovers most of the waste. The ceiling is set by how unpredictable the coin is: about 0.918 fair bits per biased flip here, or roughly 1.09 flips per fair result at best. Naming that limit is what separates a good answer from a complete one.

    Where candidates lose it

    The common loss is trying to combine flips into ranges, for example counting heads over three flips and calling some totals heads. Those schemes are almost never exactly fair with a 1/3 coin. The pair trick is exact for any bias, and saying so is the point.

    The second loss is saying two flips per fair result. You throw pairs away, so the cost is 4.5, and the interviewer will ask.

    What the interviewer asks next

    • Does the trick work if you do not know the bias?
    • How would you reuse the discarded HH and TT pairs to cut the cost?
    • How would you simulate a fair six-sided die with the same coin?

    Asked at D.E. Shaw, Research, New York, 2026 (Wall Street Oasis): How can I make an effective fair coin given a biased coin with p_heads = 1/3?

  3. 013You roll a fair die and receive Rs 100 times the face shown. Before you are paid you may reroll once, but then you must keep the second roll. What is the right to reroll worth?Expected value and decisionsCoreHedge fund long/shortMulti-manager pod

    Try it first

    Which first rolls should you keep?

    Show the worked solution

    The reroll right is worth Rs 75. A single roll is worth Rs 350 on average. With the reroll, keep 4, 5 and 6, which average Rs 500, and reroll 1, 2 and 3 for an expected Rs 350. Half the time you get Rs 500 and half the time Rs 350, so the game is worth Rs 425, Rs 75 more than without the right.

    What is the rule for keeping or rerolling?

    Think of a job offer in hand while you wait on another interview. You take the offer if it beats what you expect the other process to give you, not if it beats your dream job. Keep any result worth more than the expected value of the alternative; here the alternative is a fresh roll worth Rs 350. So 4, 5 and 6 are kept, and 1, 2 and 3 are rerolled.

    Keep any face worth more than a fresh roll, Rs 350 on averageRs 1001Rs 2002Rs 3003Rs 4004Rs 5005Rs 6006Fresh roll: Rs 350reroll: lime shows the gainkeepKeep 4, 5, 6: averageRs 500, half the timeReroll 1, 2, 3: Rs 350,half the timeGame worth Rs 425Reroll right worthRs 75
    The value of a fresh roll, Rs 350, cuts between faces 3 and 4, so you keep 4, 5 and 6 and reroll 1, 2 and 3, which makes the game worth Rs 425 and the reroll right worth Rs 75.

    How do you value the right itself?

    Value the game with the right, then subtract the game without it. With the right: half the time you hold 4, 5 or 6, averaging Rs 500; half the time you reroll and expect Rs 350. That is Rs 425. Without it the game is worth Rs 350. The right is worth the difference, Rs 75, and all of it comes from the three bad faces being replaced.

    The relationship
    V=12×500+12×350=425425−350=75V = \tfrac{1}{2}\times 500 + \tfrac{1}{2}\times 350 = 425 \qquad 425 - 350 = 75
    500average of the kept faces 4, 5 and 6, in rupees
    350expected value of a fresh roll, in rupees
    What it says in wordsThe game with the reroll right is worth the average of the kept outcomes and the expected reroll, each half the time.

    This is an option, and a research interviewer asks it to see whether you value flexibility correctly. The right has value only because you can refuse it when the first roll is good. Its value is the gain in the bad states, Rs 250, 150 and 50 on faces 1, 2 and 3, averaged over six faces: Rs 75. The same logic prices an option to expand a project or to delay an investment.

    Where candidates lose it

    The common loss is keeping only 5 and 6, or only 6, because a 4 feels ordinary. The cut-off is the expected reroll, Rs 350, and a sure Rs 400 beats it.

    The second is answering Rs 425 when asked what the right is worth. That is the game with the right; the right is the difference, Rs 75.

    What the interviewer asks next

    • What if you may reroll twice?
    • What would you pay to play if you had to announce keep or reroll before seeing the first roll?
    • How does the answer change for a 20-sided die paying Rs 100 a face?
  4. 014Estimate the annual fare revenue of one metro rail line in a large Indian city. Work from the number of stations, peak and off-peak ridership, and the average fare.Market sizing and estimationCoreIndian brokerage researchConsulting style estimation

    Try it first

    You know the line carries about 25,000 boardings in a peak hour and runs 17 hours. What goes wrong if you multiply the two?

    Show the worked solution

    About Rs 271 crore a year, on these assumptions. A weekday carries 240,000 boardings: six peak hours at 25,000, four shoulder hours at 12,000 and seven quiet hours at 6,000. Weekends and holidays run at 60% of a weekday, giving 77.5 million trips a year. At an average fare of Rs 35, that is about Rs 271 crore.

    How do you estimate a weekday's ridership?

    Think about any busy road near an office district: jammed from 8 to 11 and from 5 to 8, calm in between. A metro line has the same shape. Ridership is concentrated in two peaks, so build the day hour by hour; a flat hourly average taken from the peak overstates it badly. Assume the line runs 6 am to 11 pm, 17 hours, with 25,000 boardings in each of six peak hours, 12,000 in four shoulder hours and 6,000 in seven quiet hours. That gives 240,000.

    Weekday boardings by hour: two peaks, long quiet stretches in between0k10k20k30k6810121416182022Hour of day (24 hour clock)Morning peakEvening peaktrue average 14,118 an hourWeekday total240,000Peak rate x 17 hours425,00077% too high
    On these assumptions a weekday carries 240,000 boardings, with six peak hours at 25,000 and long quiet stretches at 6,000, so multiplying the peak rate by all 17 hours would give 425,000, about 77% too high.

    How do you check it and turn it into revenue?

    Check from the stations. A line of 25 stations at 240,000 boardings a day is about 9,600 per station, which is plausible for a mix of busy interchanges and quiet suburban stops. Two routes that land close together give you licence to use the number; if they did not, the gap would tell you which assumption to test. Then annualise: 260 weekdays at 240,000 and 105 weekend and holiday days at 60% of that give 77.52 million trips.

    StepAssumptionResult
    Weekday boardings6 x 25,000 + 4 x 12,000 + 7 x 6,000240,000
    Weekday trips a year260 days62.4 m
    Weekend and holiday trips105 days at 60%15.12 m
    Average fareDistance-based, blendedRs 35
    Annual fare revenueRs 271 crore
    Every input is an assumption for the exercise; the structure matters more than any single number.

    Say which input you trust least: the average fare, since metro fares usually rise with distance and a line's trip-length mix is hard to guess. A research analyst would also note that fare revenue is only part of a metro's income, with advertising and property often material, and would check the operator's published ridership rather than rely on the estimate.

    Where candidates lose it

    The trap is taking a peak-hour number and multiplying by operating hours, which inflates the day by more than 75% on these assumptions. The interviewer is listening for whether you think about the shape of demand through the day.

    The second is giving one number with no check. Tie the daily figure back to the station count, and name the fare as your softest assumption.

    What the interviewer asks next

    • How would a new interchange with another line change the estimate?
    • What fare increase would offset a 10% fall in ridership?
    • How would you estimate the line's non-fare revenue?
  5. 015A fund earns 12% a year before fees for 20 years and charges a 2% annual fee. How much of the final wealth does the fee take?Returns and compoundingCoreLong-only asset managementResearch KPO and GCC

    Try it first

    Guess first: what share of the gross ending wealth does the 2% fee take after 20 years?

    Show the worked solution

    About 30% of the final wealth. At 12% a rupee grows to 9.65 in 20 years; at 10% after fees it grows to 6.73. The difference, 2.92, is 30.3% of the gross result. A fee that looks like one sixth of the annual return takes close to a third of the ending wealth, because the fee compounds too.

    Why is the answer so much bigger than 2%?

    Think of a leaking water tank that loses a small share of its contents every day. The leak looks trivial against the day's inflow, but it runs every day on the whole tank, and over a year it drains a large share of what would have collected. A percentage fee is charged on the whole balance every year, and every rupee it removes also loses the growth it would have earned for the rest of the period.

    A 2% fee each year takes almost a third of what 12% would have built in 20 years0x2x4x6x8x10xYear 0Year 5Year 10Year 15Year 2012% gross: 9.65x10% net: 6.73xFee takes 2.92x30% of the gross2 points a year looks like one sixth of 12%,but by year 20 it is 30% of the wealth
    Over 20 years a rupee grows to 9.65 at 12% gross but only 6.73 at 10% after a 2% annual fee, so the fee takes 2.92, or 30% of the gross ending wealth.

    How do you work it quickly in the room?

    Use the ratio rather than the two big numbers. The net path grows at 1.10 against 1.12, so each year it keeps 1.10 / 1.12 = 98.2% of the gross path. Over 20 years that ratio compounds to 0.982 to the 20th, about 0.70, so the fee takes about 30%. A mental shortcut: 20 years at roughly 1.8% a year is about 36% by simple addition, and compounding pulls it back to about 30%.

    The relationship
    1−(1.101.12)20=1−6.7279.646=30.3%1 - \left(\frac{1.10}{1.12}\right)^{20} = 1 - \frac{6.727}{9.646} = 30.3\%
    1.10one year of growth after the fee
    1.12one year of growth before the fee
    20years
    What it says in wordsThe share of wealth the fee takes is one minus the net-to-gross ratio compounded over the period.

    The same arithmetic applies to any recurring drag: fund expenses, trading costs, a tax on annual gains. It is also why a long-horizon investor compares costs in basis points. The limitation: the calculation assumes the gross return is the same with and without the fee; a manager who earns the fee through better returns changes the comparison.

    Where candidates lose it

    The common loss is answering 2% or 40%, either treating the fee as one-off or multiplying 2% by 20 years. Both skip compounding, which runs on the fee as well as on the return.

    The second is calculating the two multiples correctly and then dividing the gap by the net result rather than the gross. The question asks what share of the gross result the fee takes.

    What the interviewer asks next

    • What fee would take half the gross wealth over 30 years?
    • How does the answer change if the gross return is 8% instead of 12%?
    • What return would an active fund need before fees to match a 0.2% fee index fund returning 12% gross?
  6. 016A stock trades at Rs 1,000. It generated free cash flow of Rs 30 a share this year and its cost of equity is 11%. If the cash flow grows at a constant rate forever, what growth rate is the market assuming?Valuation riddlesCoreBuy-side equity researchHedge fund long/short

    Try it first

    Quick estimate first: roughly what growth is priced in?

    Show the worked solution

    About 7.8% a year, forever. In a growing perpetuity the price equals next year's cash flow, Rs 30 x (1 + g), divided by 11% minus g. Setting that equal to Rs 1,000 and solving gives g = (110 - 30) / 1,030 = 7.77%. The quick version: an 11% required return minus a 3% cash yield leaves about 8% for growth.

    Why solve for growth instead of assuming it?

    Think of a shop listed for sale at a price that looks high for its current takings. Rather than argue about the price, ask what sales growth would justify it; then the argument becomes whether that growth is believable. A price is a bundle of assumptions, and solving for the growth it needs turns a vague sense that a stock is expensive into one number you can test. This is the idea behind a reverse DCFA valuation run backwards: start from the market price and solve for the growth or margins the price implies..

    Solve for the growth the price needs: the Rs 1,000 line crosses at 7.8%Rs 0Rs 500Rs 1,000Rs 1,500Rs 2,0000%2%4%6%8%10%Perpetual growth assumedShare price Rs 1,0007.8%7%: Rs 8038.5%: Rs 1,302Price = FCF x (1+g)divided by (11% - g)Solve for g:(1,000 x 11% - 30)/ (1,000 + 30)g = 7.77%
    With Rs 30 of free cash flow and an 11% cost of equity, value rises steeply with the growth assumed and reaches the Rs 1,000 share price at 7.8% perpetual growth; at 7% the value would be only Rs 803.

    How do you solve it cleanly?

    Write the growing perpetuity with next year's cash flow on top: 1,000 = 30 x (1 + g) / (0.11 - g). Multiply out: 110 - 1,000g = 30 + 30g, so 80 = 1,030g and g = 7.77%. The shortcut, cost of equity minus cash yield, gives 8% and is close because the yield is small; the exact answer is a little lower because the Rs 30 grows before it is paid.

    The relationship
    g=P⋅r−FCF0P+FCF0=1,000×0.11−301,000+30=7.77%g = \frac{P \cdot r - FCF_0}{P + FCF_0} = \frac{1{,}000 \times 0.11 - 30}{1{,}000 + 30} = 7.77\%
    Pthe share price, Rs 1,000
    rthe cost of equity, 11%
    FCF_0this year's free cash flow a share, Rs 30
    What it says in wordsThe implied growth is the part of the required return that the current cash yield does not supply.

    Then judge it. Growing at 7.8% forever means growing faster than most economies can for ever, which is a demanding assumption. Look at how steep the curve is near the answer: at 7% the stock would be worth Rs 803, at 8.5% Rs 1,302. Small changes in the growth belief move the value a lot, which is why a single-stage model is a sense check here, not a valuation.

    Where candidates lose it

    The common loss is dividing 30 by 1,000 and answering 3%, confusing the cash yield with growth. The required return is the yield plus growth, not the yield alone.

    The second is stopping at 8% without noting it is approximate, or quoting 7.8% without judging whether perpetual growth at that rate is plausible. The interviewer wants the number and a view on it.

    What the interviewer asks next

    • What cost of equity would make 5% growth consistent with the price?
    • How would you run the same test with a two-stage model?
    • What would you check in the accounts to see whether 7.8% growth is achievable?
  7. 017A company reported net income of Rs 100 crore, depreciation of Rs 20 crore and capex of Rs 20 crore. There were no debt or dividend movements, yet its cash fell by Rs 30 crore. Receivables rose Rs 80 crore, inventory rose Rs 60 crore and payables rose Rs 10 crore. Reconcile the numbers.Three statement riddlesCoreSell-side equity researchResearch KPO and GCC

    Try it first

    What was operating cash flow?

    Show the worked solution

    Working capital absorbed Rs 130 crore, more than all of the profit. Start at net income of 100 and add depreciation of 20. Receivables up 80 and inventory up 60 each tie up cash; payables up 10 frees some. Operating cash flow is minus 10. Capex of 20 takes the change in cash to minus 30, matching the fall.

    How can a profitable company lose cash?

    Think of a wholesaler who sells a lot this month, but on 90 days' credit, and stocks up for the festive season. The books show a profit, yet the bank balance falls, because the sales have not been collected and the new stock has been paid for. Profit counts sales when they are made; cash counts them when they are collected, and growing receivables and inventory are the gap between the two.

    From Rs 100 crore of profit to Rs 30 crore less cash, Rs crore0100Netincome+20D&A-80Receivablesup-60Inventoryup+10Payablesup-20Capex-30CashchangeWorking capital absorbs Rs 130 croreOperating cashflow -10
    Rs 100 crore of net income plus Rs 20 crore of depreciation is more than consumed by Rs 80 crore of new receivables and Rs 60 crore of new inventory, leaving operating cash flow of Rs -10 crore and a Rs 30 crore fall in cash after Rs 20 crore of capex.

    Which way does each working capital line push cash?

    Read the balance sheet change and ask whether cash went out or came in. A rise in an asset such as receivables or inventory uses cash; a rise in a liability such as payables provides it. Receivables and inventory together took Rs 140 crore and payables returned Rs 10 crore, so working capital absorbed Rs 130 crore against Rs 120 crore of profit plus depreciation. That makes operating cash flow minus 10.

    LineRs croreWhy
    Net income100starting point
    Depreciation+20non-cash charge added back
    Receivables up-80sales not yet collected
    Inventory up-60stock bought, not yet sold
    Payables up+10suppliers not yet paid
    Operating cash flow-10
    Capex-20investment in assets
    Change in cash-30
    The indirect cash flow statement, which starts from profit and walks each balance sheet change back to cash.

    What a research analyst does next: compare the rise in receivables to sales growth. If receivables grew much faster than revenue, customers are paying more slowly or sales were pushed through on easy terms, and that is a question for management. The limitation: a fast-growing company can have this pattern for healthy reasons, so the answer needs the trend, not one year.

    Where candidates lose it

    The common loss is getting the working capital signs backwards and adding the rise in receivables. An asset going up is cash going out; say that rule before you calculate.

    The second is reconciling the arithmetic and stopping. The interviewer also wants a view: receivables up 80 on profit of 100 is a flag worth naming.

    What the interviewer asks next

    • What would you want to know about revenue growth before judging the receivables increase?
    • How would days sales outstanding help here?
    • If the company then factored its receivables for cash, how would the statements change?
  8. 018A company's pre-tax profit is flat year on year, but its effective tax rate falls from 30% to 25%. How much does EPS grow, and should the stock's P/E multiple rise because of it?EPS and share countCoreSell-side equity researchHedge fund long/short

    Try it first

    By how much does EPS grow, with the share count unchanged?

    Show the worked solution

    EPS grows 7.1%, and the multiple should not rise on that growth. Net income goes from 70 to 75 on flat pre-tax profit of 100. The price can reasonably rise about 7.1% at the same P/E, because each share now earns more. But the growth happens once: next year, with pre-tax profit still flat, EPS growth is zero.

    Where does the 7.1% come from?

    Think of a salaried employee whose tax bill falls. Take-home pay jumps once, but next year's take-home pay only grows if the salary does. A lower tax rate raises net income by the tax saved, here 5 on a base of 70, which is 7.1%, even though the business earned nothing more. Say the base out loud: people who answer 5% are measuring against pre-tax profit.

    A lower tax rate lifts earnings once; it does not raise the growth rateNet income 70Tax 30Tax 30%Net income 75Tax 25Tax 25%Pre-tax profit 100 in both yearsEarnings growth by year+7.1%Year 10%Year 20%Year 3The rate is already in the base after year 1:a one-off step, not faster growth
    Pre-tax profit stays at 100 while tax falls from 30 to 25, so net income rises from 70 to 75, a 7.1% step in year one followed by zero growth in later years because the lower rate is already in the base.

    Why should the multiple stay where it is?

    A P/E multiple pays for the level of earnings and for their future growth. The tax cut raises the level, so at an unchanged P/E of 20 the price rises 7.1% to match. It does not raise future growth, so a higher multiple would pay twice for the same one-off step. Re-rate to 22x on the back of 7% growth and the price would rise 17.9%, most of it paying for growth that will not repeat.

    Two caveats a research analyst adds. First, ask why the rate fell: a statutory cut is durable, while a one-time credit or a shift of profit to a lower-tax region may reverse, in which case even the level change deserves a discount. Second, check whether competitors got the same cut; if they did, some of it may be passed to customers through prices. Confirm any tax rate you use against the current rules rather than memory.

    Where candidates lose it

    The trap is treating the EPS jump like organic growth and arguing for a higher multiple because earnings grew faster. Growth that comes from a rate change is a step, and a step is paid for once.

    The second loss is answering 5%, the change in the tax rate, instead of 7.1%, the change in net income.

    What the interviewer asks next

    • What if the lower rate comes from a one-time tax credit?
    • How would you show this in an EPS bridge from last year to this year?
    • How does the answer change if the company also buys back 5% of its shares?
  9. 019A company's bond has a modified duration of 6. Its credit spread widens by 150 basis points while government yields stay put. Roughly how much does the bond's price fall, and what does the move tell an equity research analyst about the company's cost of debt?Cost of capital and ratesCoreAQR Capital ManagementGreenwich · 2021

    Try it first

    Roughly how far does the bond price fall?

    Show the worked solution

    About 9%: duration of 6 times a 1.5 point rise in yield. For an 8-year, 7% coupon bond at par, the true fall is 8.5%, a little less than the straight-line estimate because the price curve bends. For the equity analyst, the market now charges the company about 1.5 points more for new debt, so the cost of debt in WACC should use the new yield, not the old coupon.

    How does duration turn a spread move into a price move?

    Think of a fixed-rate deposit you cannot break. When new deposits start paying more, yours is worth less to anyone who might buy it from you, and the longer it has left to run, the bigger the discount. Modified duration is that sensitivity in one number: the approximate percentage price change for each 1 point change in yield. A spread widening raises the bond's yield by the same amount, so 6 x 1.5 = 9%.

    Duration x yield change gives the price move; the curve is gentler for big moves7080901001101204%6%7%8.5%10%Yield+150 bpPar, 100True price 91.5: -8.5%Duration estimate 91.0: -9.0%Tangent: theduration lineCurve: thereal price
    For an 8-year, 7% coupon bond priced at par with modified duration 5.97, the duration tangent predicts a 9.0% fall for a 150 basis point rise in yield, while the true price falls 8.5%, because the price curve bends away from the straight line.

    Why is the true fall a little smaller than 9%?

    The price-yield curve is convex: it flattens as yields rise. Duration draws a straight tangent at today's yield, so for large moves it overstates price falls and understates price rises. Here the tangent says 91.0 and the bond is actually worth 91.5. For small moves the gap is negligible; at 150 basis points it is about 0.5 points, worth one sentence.

    The relationship
    ΔPP≈−Dmod Δy=−6×0.015=−9%\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y = -6 \times 0.015 = -9\%
    D_modmodified duration, 6
    Δythe change in yield, here the 1.5 point spread widening
    What it says in wordsThe percentage price change is roughly duration times the yield change, with the sign reversed.

    Now the equity view. The cost of debt in a WACC is what the company would pay to borrow today, so a 150 basis point widening raises it by about 1.5 points, whatever the coupon on existing bonds. With debt at 30% of capital and a 25% tax rate, that alone adds about 0.34 points to WACC. The larger message is the signal: credit investors are pricing more risk, and equity sits below the bonds, so the analyst should ask what the bond market has seen.

    Where candidates lose it

    The common loss is confusing basis points and percentage points and answering 900% or 0.9%. Say that 150 basis points is 1.5 points before multiplying.

    The second is keeping the old coupon as the cost of debt in the WACC. A coupon set years ago is history; the market yield today is what new debt would cost.

    What the interviewer asks next

    • What would the same widening do to a 2-year bond from the same company?
    • Why might the stock fall by more than the bond in this situation?
    • How would you estimate the effect on the company's interest cover when the debt is refinanced?

    Asked at AQR Capital Management, Investment Research, Greenwich, 2021 (Wall Street Oasis): Discussion on credit spreads on fixed income products and duration.

  10. 020A company's revenue grew 12% and its prices rose 5%. How much did volume grow?Growth, mix and unit economicsWarm upSell-side equity researchIndian brokerage research

    Try it first

    Answer in five seconds.

    Show the worked solution

    About 6.7%. Revenue is price times volume, so the growth factors multiply: 1.12 = 1.05 x (1 + v). Dividing, 1.12 / 1.05 = 1.0667, so volume grew about 6.67%. Subtracting 5 from 12 gives 7%, which overstates volume by the small cross term, 0.33 points.

    Why divide instead of subtract?

    Think of a tea stall that raises the price of a cup and also sells more cups. The takings grow for both reasons, and the extra cups are also sold at the higher price, which is a little extra neither change produces alone. Revenue is price times volume, so growth factors multiply, and the way to back one out is to divide. 1.12 divided by 1.05 is 1.0667.

    Revenue is price x volume, so growth rates multiply: divide, do not subtractLast year's revenueprice 1.00 x volume 1.00Volume +6.67% at the old pricePrice: 1.00, then 1.05Price +5% onlast year's volumeCorner: 5% x 6.67%= 0.33%1.12 / 1.05= 1.0667
    Drawing revenue as price times volume, a 5% wider and 6.67% taller rectangle has 12% more area: a 5% price strip, a 6.67% volume strip and a 0.33% corner where the extra volume is sold at the higher price.

    When does the shortcut of subtracting matter?

    The error from subtracting is the corner of the rectangle, price growth times volume growth. At single-digit rates the corner is small, about 0.33 points here, but it grows quickly with the rates, and in high-inflation markets it becomes the whole story. With 30% price rises and 40% revenue growth, subtracting says 10% volume growth; dividing says 7.7%.

    The relationship
    1+v=1+grev1+gprice=1.121.05=1.06671 + v = \frac{1 + g_{rev}}{1 + g_{price}} = \frac{1.12}{1.05} = 1.0667
    vvolume growth
    g_revrevenue growth, 12%
    g_priceprice growth, 5%
    What it says in wordsVolume growth is revenue growth divided by price growth, as factors.

    For an analyst this split is the first question on any results call: how much of the growth was price and how much was volume. Volume growth says whether the company is winning customers; price growth says whether it has pricing power or is passing on costs. The limitation: reported price growth often mixes true price changes with shifts in product mix, so ask how the company defines it.

    Where candidates lose it

    The trap is subtracting and answering 7% without a second thought. It is close enough to pass as mental arithmetic but wrong in method, and the interviewer asks the question to hear which one you use.

    Say the division, give 6.7%, and add that the gap is the cross term. That turns a one-line answer into evidence of method.

    What the interviewer asks next

    • Revenue fell 3% while prices rose 8%. What happened to volume?
    • How would you split growth into price, volume and mix for a company selling three products?
    • Why might management prefer to report volume growth before or after the effect of mix?